Large N correlators of Chern-Simons theory in lens spaces

We compute the invariants for a class of knots and links in arbitrary representations in S-3/Z(p) in the large k (level), large N (rank) limit, keeping N/(k+N) = lambda fixed, in U(N) and Sp(N) Chern-Simons theories. Using the relation between the saddle-point description and collective field theory, we first find that the invariants for the Hopf link and unknot are given by the on-shell collective field theory action. We next show that the results of these two invariants can be used to compute the invariants of other torus knots and links. We also discuss the large N phase structure of the Hopf link invariant and observe that the same may admit a Douglas-Kazakov type phase transition depending on the choice of representations and lambda.

Automated summary

In this study, invariants for a class of knots and links in arbitrary representations in S-3/Z(p) were computed in the large k, large N limit in U(N) and Sp(N) Chern-Simons theories. The results for the Hopf link and unknot invariants were found to be related to collective field theory, and were used to compute invariants for other torus knots and links. The phase structure of the Hopf link invariant in the large N limit was discussed, indicating a possible phase transition depending on representations and lambda.